Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression
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This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity \(r=2\), a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.
KeywordsMultiwavelet Biorthogonal multiwavelet Analysis-ready multiwavelet Transferring armlet order
Mathematics Subject Classification42C40a 65T60 15A23 94A08
The authors thank anonymous reviewers and the editor-in-chief, Prof. M.N.S.Swamy, for their valuable suggestions and comments for improving the presentation of this paper. This work was supported in part by NSFC under Grant No. 11301504 and in part by the President Fund of University of Chinese Academy of Sciences under Grant No. Y25101HY00.
- 3.H. Bray, K. McCormick, R.O. Wells, X. Zhou, Wavelet variations on the Shannon sampling theorem. Curr. Mod. Biol. 34(1–3), 249–257 (1995)Google Scholar
- 7.T.N.T. Goodman, C.A. Micchelli, Orthonormal Cardinal Functions, in Wavelets: Theory, Algorithms, and Applications (Academic, San Diego, CA, 1994)Google Scholar
- 10.Q.T. Jiang, Orthogonal and biorthogonal square-root(3)-refinement wavelets for hexagonal data processing. IEEE Trans. Signal Process. 57(11), 4313–14304 (2009)Google Scholar
- 12.J. Lebrun, M. vetterli, Balanced multiwavelets, IEEE international conference on acoustics, speech and signal processing, vol. 3 (1997), pp. 2473–2476Google Scholar
- 19.L. Liu, H. Zhang, Application on Image fusion based on balanced multi-wavelet, 2010 International Symposium on Intelligence Information Processing and Trusted Computing, 512–515 (2010)Google Scholar
- 21.Walid A. Mahmoud, Majed E. Alneby, Wael H. Zayer, 2D-multiwavelet transform 2D-two activation function wavelet network based face recognition. J. Appl. Sci. Res. 6(8), 1019–1028 (2010)Google Scholar
- 30.P.P. Vaidyanathan, Multirate Systems and Filter Banks, Englewood Cliffs (Prentice Hall, NJ, 1993)Google Scholar
- 32.C. Weidmann, J. Lebrun, M. Vetterli, Significance tree image coding using balanced multiwavelets. Proc. ICIP, Chicago, IL, Oct. 1, 97–101 (1998)Google Scholar
- 34.X.-G. Xia, Z. Zhang, On sampling theorem, wavelets, and wavelet transforms. IEEE Trans. Signal Process. 41(12), 2535–3524 (1993)Google Scholar